It can mean contest (verb)as in "I challenge that decision" - I do not agree with the decision and I want it to be reviewed or action to be taken to change the decision.

It can also be used as in "I challenge you to a fight" - I propose that we fight (with the implication that it is a risky/dangerous undertaking and that the other person will 'lose face' if they don't agree to the challenge).

There are very probably other shades of meaning as well, but it would not be used as a synonym for try, as in your example above.